Associating significance with a dodecahedron

Time for Provocative Mnemonic Aids to Systemic Connectivity? (Part #9)

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As discussed separately, efforts are made to associate logical significance with a variety of polyhedra (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018). Polyhedra may be used for experimental mappings of disparate concepts of which their use for configuring articles of various human rights charters are most relevant to the above argument (Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008). In this respect it is appropriate to stress again that such sets of values are only otherwise configured as checklists lacking ny systemic organization. Indeed it could be asked how fundamental human values are organized other than in that manner -- if there is any consensus on what such values are.

The speculative challenge is whether the considerable interest in the pattern of the Zodiac, given the value-related meanings held to be associated with it, suggests the possibility of some such mapping, notably in the light of the preliminary exploration of Paul Schatz. One approach to the development of the argument gave rise to a variant of the image on the left below, in the light of the work on the Geometry of Meaning (1976) of Arthur Young, aa described separately (Geometry of meaning: an alchemical Rosetta Stone? 2013). This concluded a discussion of Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships (2013).

The pattern of the image on the left below can be seen as framing the question as to whether it can be understood as a 2D projection of a form in 3D (or more). In particular does its 12-foldness relate in any way to that of the cube -- especially as inverted by Schatz (as indicated above).

In the absence of any other indication as to how a set of 12 concepts or functions might be mapped onto a dodecahedron (as one logical option), use was made of an early dice which had such a mapping of signs of the Zodiac (Silberbeschlagener Würfel mit den Namen der Sternzeichen, gefunden Genf, 1982). This gave rise to the mapping in the image and animation (centre and right below). Images of 12-sided dice used for various purposes are offered by Wikipedia, although these do not include any with the Zodiac. There is some discussion of Board games with dodecahedral dice. Use of dodecahedral dice in relation to Dungeous and Dragons is the subject of an extended commentary.

The centre image is a development of that of Schatz (as reproduced above, to which the colours below were added). It derives from unwrapping the 3D dodecahedron into a 2D net. Of particular interest is that the similarly coloured pentagons correspond to opposite positions in the traditional configuration image on the left (to which the corresponding "measure formulae" have been added according to the argument of Arthur Young). He presented that configuration of learning cycles -- as a Rosetta stone of meaning.

Traditional pattern of Zodiac with associated "measure formulae"	Zodiac signs on dodecahedral net according to dice attribution	Animation of dodecahedron with Zodiac signs from dice
Following the attributions of Arthur Young (1976)	Animations generated using Stella Polyhedron Navigator

Curiously the signs which are opposite in the image on the left are contiguous as coloured in the central pattern (when folded into the 3D form on the right). There is however one exceptional set of opposites -- coloured turquoise. Although opposite in the image on the left, they are on opposite sides of the 3D version and are therefore not contiguous.

Sociophysics? Young's amendment highlights a progression in temporal reciprocation (or inversion) which features in an appropriately titled subsequent study (Nested Time, 2004). He successively distinguishes: production capacity, is effectively timeless as a product of 1/T⁰; change over time as a product of 1/T¹ (or T^-1); rate of change as a product of 1/T² (or T^-2); and a measure of control as a product of 1/T³ (or T^--3).

This could now be explored within the controversial framework of sociophysics (divisive in its own right), usefully summarized by Frank Schweitzer (Sociophysics, Physics Today, 71, 2018). A commentary with respect to Young's presentation features in the work of Paris Arnopoulos (Sociophysics: Cosmos and Chaos in Nature and Culture, 1993):

Since power is the rate of applying force, controlling this rate is of utmost importance. Control has been identified as the capacity to modify the rate of change, ie to speed it up or slow it down. Therefore, power control is a necessary ingredient of any orderly social change. The mathematical definition of power, and its algebraic equivalents show that:

P = W/t = Fv = ma(s/t) = m(s/t²)(s/t) = ms(s/t³) = msc

This last parenthesis (s/t³) has been defined by Young as control (c), and translates as the rate of change of acceleration. It will be recalled that since v=s/t and a=s/t², control becomes the third derivative of velocity....

Since power is directly proportional to the rate of energy conversion or information flow, dynamic systems require a great degree of control. As people become more energetic or informed, they tend to get out of control; so in order to avoid that, dynamic societies must become more regulated. It may therefore be said that the kind of government that a system has depends on the amount of power it disposes. (p. 82)

He develops the argument otherwise as a means of engaging with the Triple Helix model of innovation thesis (Braiding the Triadic Codex and Triple Helix: the sociophysics of nature-culture-nurture and academy-industry-polity, 2000). There he notes:

... this short paper interfaces with the triple helix paradigm by weaving its triadic social focus-locus with the power-wealth-data flows among its state-market-school centers. In this way we can concentrate on the most significant influence-finance-science transactions of the polity-industry-academy triangle.... In doing any job, force performs work: W = Fs =mas= mv2. This means that some work must be done in order to bring about social change. If that change is needed fast then one must exert a lot of power: P = W/t =mav = Fv. By this mathematical transformation, we have arrived at this crucial notion of power politics as well as physics. Social power however, unlike physical power, does not move inanimate objects but human masses to act far and fast.... Informative societies are negentropic because they increase systemic organization and decrease environmental degradation. Accumulating human knowledge also improves social control (C = a/t), since it regulates social change in a more enlightened manner. For that reason the exercise of responsible
social power requires strict political control (P = msC)....

Unusually Arnopoulos presents a synoptic overview in schematic form (p. 84) of the interrelationship between 15 fundamental concepts deriving from a triadic hypothesis (space-time-existence) correlating space curvature, material density and universal time (p. 5).

These suggest an interesting relationship to Young's 12-fold Rosetta stone of meaning, as depicted above (especially if 3 are omitted or conflated in some way). Otherwise it would take the form of three pentagons, thereby suggestive of arguments in relation to the Chinese understanding of the 5-fold Wu Xing (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012) and to the 15 transformations of Christopher Alexander, as discussed separately (Tentative adaptation of Alexander's 15 transformations to the psychosocial realm, 2010).

Tentative preliminary amendment of the dodecahedral configuration

using both Arthur Young (1976) and Paris Arnopoulos (2000)

Of relevance to the triadic hypothesis articulated by Arnopoulos is the subsequent argument of T.N. Palmer (The Invariant Set Hypothesis: a new geometric framework for the foundations of quantum theory and the role played by gravity, Electronic Notes in Theoretical Computer Science, 2011).

Also calling for integration with such insights is the extensive work on systematics by John Bennett (The Dramatic Universe, 1955-66), as introduced by Anthony Blake (Systematics, Duversity), notably with respect to 12-fold sets (Overview of 12 Systems).

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